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Section 6.2 Worksheets

PDF Version of these Worksheets

Worksheet Worksheet 1
A4US

1.

Calculate \(t^2 \cdot t^4\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result.

2.

Calculate \(\left( a^3 \right)^4\) using a presentation that shows all of the individual steps. Then verify that the power rule gives the same result.

3.

Calculate \(y^{-3} \cdot y^5\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result.

Worksheet Worksheet 2
A4US

1.

Calculate \(x^{-2} \cdot x^{-3}\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result.

2.

Calculate \(\left( x^{-1} \right)^{-n}\) using the power rule. Then rewrite the part of the expression inside the parentheses using the definition of negative exponents. In order for the math to be consistent, the two results should be equal. Explain how this verifies the first formula in Theorem 6.1.4.

3.

Start from the equation \(x^{-n} = \frac{1}{x^n}\) and take the reciprocal of both sides of the equation. Explain how this verifies the second formula in Theorem 6.1.4.

Worksheet Worksheet 3
A4US

1.

Calculate \(x^{2} \cdot x^{-5}\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result. Give your final answer in the form \(x^n\) for some number \(n\text{.}\)

2.

Calculate \(x^{2n} \cdot x^{3n}\) using the product rule. Explain the logic of your result in complete sentences.

3.

Consider the following presentation:

\begin{equation*} \begin{aligned} x^3 \cdot x^{-3} \amp = x^3 \cdot \frac{1}{x^3} \amp \eqnspacer \amp \text{Definition of negative exponents} \\ \amp = \frac{x^3}{x^3} \amp \amp \text{Multiply fractions} \\ \amp = \frac{x \cdot x \cdot x}{x \cdot x \cdot x} \amp \amp \text{Definition of exponents} \\ \amp = \frac{\cancel{x \cdot x \cdot x}}{\cancel{x \cdot x \cdot x}} \amp \amp \text{Reduce the fraction} \\ \amp = 0 \end{aligned} \end{equation*}

Identify and explain the error. What would you suggest as a way for students to avoid this mistake?

Worksheet Worksheet 4
A4US

1.

Calculate \(\left( x^{3} \right)^{-4}\) using a presentation that shows all of the individual steps. Then verify that the power rule gives the same result. Give your final answer in the form \(x^n\) for some number \(n\text{.}\)

2.

Calculate \(\left( x^{2m} \right)^{3n}\) using the power rule. Explain the logic of your result in complete sentences.

3.

Calculate \(x^{4} \cdot x^{-4}\) using a presentation that shows all of the individual steps. Then verify that the product rule gives the same result.

Worksheet Worksheet 5
A4US

1.

Calculate \(\left( x^{-3} \right)^{-4}\) using a presentation that shows all of the individual steps. Then verify that the power rule gives the same result. Give your final answer in the form \(x^n\) for some number \(n\text{.}\)

2.

Calculate \(x^n \cdot x^m \cdot x^p\text{.}\) Explain the logic of your result.

3.

Calculate \(\left( \left( x^n \right)^m \right)^p\text{.}\) Explain the logic of your result.